It is generally appreciated by voltmeter users that the garden variety AC voltmeter which is "average responding but calibrated in RMS volts" gives correct indications only for sinusoids, and only then provided the input is a single frequency signal free of significant distortion. While these constraints can sometimes be met, they frequently can not be met or are simply ignored. Thus, on the one hand commercially available DC voltmeters routinely measure one volt out to six or eight places; while on the other hand, the typical AC measurement made with an average responding voltmeter can't be trusted to more than four places. For this and other reasons there has been increasing interest in AC voltmeters that respond directly to the RMS value of the applied signal instead of to the average value.
The essential ingredient of an RMS meter is the RMS converter. One type of RMS converter produces a DC output voltage that is proportional to the RMS value of an applied input that is itself either DC or an AC signal of components above a certain frequency and below another. Thermal converters and analog logging circuits are frequently used as RMS converters. Techniques also are in use where the incoming waveform is periodically accurately sampled, digitized and the RMS value computed. The Hewlett-Packard 3458A is an example of this latter technique.
One might be tempted to say that the community of users for voltmeters expects the accuracy of AC measurements to improve commensurate with the ongoing improvements in DC measurement. This suggests an increased interest in RMS measurements, and indeed, many RMS converters are now on the market: the cost of RMS measurement is coming down and its typical accuracy is going up.
Nevertheless, the measurement of the true RMS voltage of a time varying signal is frequently more difficult than is ordinarily appreciated. Consider the following three aspects of RMS measurement.
The first of these aspects is selection of the time constant for RMS converter response. Consider the measurement of the RMS voltage of an arbitrary waveform. Suppose this particular waveform represents some kind of modulation that comes and goes. When there is no modulation we'd prefer to say that the RMS value of the modulation signal is zero. When the modulation is present it has some apparent value. Now suppose the modulation is continuously present thirty percent of the time and absent the remaining seventy percent of the time. Should it make a difference if the on/off times are thirty/seventy seconds as opposed to thirty/seventy milliseconds?. Or microseconds?. Most users would expect that it would make a difference, and indeed would want it to. It will make a difference if what is desired is the RMS value of the modulation when it is present (and excluding its absence), as compared to measuring the overall value for all modulation as an ongoing whole. Some users may desire a time constant (or averaging window) of several seconds, or even several tens of seconds, while others would be more conveniently served by a time constant of a few tenths of a second.
RMS stands for Root Mean Squared, and what our example illustrates is that the Mean or averaging part of the process has some way diminishing the contribution of past instances of signal values. There are at least three ways this can be done: exponential decay with a time constant (ala thermal converters); travelling averages; and, simple averages over a batch of readings taken within a specified time interval.
This diminution of the contribution of past events in the signal is desirable, if for no other reason, so that the reading on the voltmeter "comes up" (settles) within an "appropriate" length of time after application of the signal to the meter, and that it thus also similarly reflects changes in the applied value considered to be "steady state" changes.
To belabor the modulation signal example a bit further, it is clear that the considerations that surround the "duty cycle" issue of signal on versus signal off discussed above also attach themselves to what goes on while the signal is on. That is, to what extent is a short but extreme increase or decrease in the signal level to affect, and then persist in, the reading?. At root, the duty cycle issue and the response time issue are really the same thing, viewed from different perspectives.
Clearly then, a general purpose RMS meter ought to have a selectable response time. Such selectability is easier said than done, however. How does one change the (thermal) time constant of a thermal converter constructed of thermocouples isolated in an evacuated glass envelope?. It is true that the time constants in logging circuit type RMS converters can be changed. However, such switching (probably of premium grade capacitors exhibiting low dielectric absorption) is not without additional expense, and offers only preselected choices. The digitize and compute type of RMS converter would appear to have an advantage in this respect, since it employs digital filtering that does not make use of actual switches and actual capacitors.
The second aspect of interest is that of low frequency tracking. It is related to the notion of the time constant for response, in that at any point in time an RMS converter will have some particular (and therefore finite) time constant, even if it was selected from among a large number of possibilities. What low frequency tracking means is that for any given time constant, no matter how long, there will be input frequencies that the RMS converter treats as slowly changing steady state signals. That is, for those signals the output of the RMS converter will be a slowly changing output separable into a steady state (DC) component and a changing (AC) component. The lower the input frequency gets the smaller the DC output component gets and the larger the AC component gets. A thermal converter measuring 1 Hz would exhibit this behavior. At 0.1 Hz the situation is even worse, and would be rationalized by saying: "Well, it works great at DC, and this is just a varying DC input." It is important to recognize that this situation applies to all RMS converters, and not just to thermal converters. It happens because there must be a definite time constant.
The effects of tracking at high or medium frequencies are generally alleviated by filtering the output of the RMS converter. In principle such filtering introduces an error (as shown below), but in practice the error is relatively small and is considered a small price to pay to avoid an annoying jitter in the measurement.
Low frequency tracking can be a bothersome situation. On the one hand, if left unfiltered it produces an uncertainty or jitter in the displayed result, and although the variation in that result might warn the user of the uncertainty there is no guarantee that he will interpret the jitter correctly He might conclude that the signal is noisy. In any event, experience suggests that many users prefer stable readings, even at the expense of some inaccuracy. But on the other hand, heavy filtering of the output of the RMS converter ahead of the actual measurement circuit that is sufficient to conceal the presence of low frequency tracking will introduce significant errors in the result.
To appreciate why such filtering produces an error, consider the following case. Suppose a low frequency (say, one hertz) two volts peak-to-peak sine wave is applied to an RMS converter with a time constant that is relatively short compared to the period of th applied input, say a time constant of one twentieth or one fiftieth of a second. For the sake of analysis, further assume that the gain of the RMS converter is unity, so that for one volt RMS in we are to expect one volt RMS out. (In our example, the RMS input is 0.707 volts.) What does the output of this RMS converter look like under these conditions?.
First, the output closely resembles an absolute value function performed upon the input, much as the (unfiltered) output of a bridge rectifier in a power supply is related to a sinusoidal input. It is close to that, although not exactly. For whereas the input to the RMS converter passes through zero twice a second, the output of the RMS converter never falls completely to zero. For that to happen the input would have to remain at zero for some definite length of time, which would be on the order of five to ten time constants. Similarly, the output never quite rises to one volt, although it would get pretty close.
Now filter the output, say with an RC low-pass filter. The answer is, of course, between zero and one volt, and is probably not too far from 0.6366 volts, which is the familiar constant describing the relationship between a sine wave's peak value and its average rectified value. But any deviation from a value of 0.707 volts is an error. Now, the lower the frequency the more the converter's output resembles the output of a bridge rectifier, and the more the answer approaches the limiting value of 0.6366 volts. Clearly one can't simply filter the output of the RMS converter and restore its output to a DC value that is the same as the RMS value of the input. To do that, one must take the RMS value of the output, which requires combining the RMS value of the AC component with the DC component. It boils down to this: As tracking gets worse, more of the RMS converter's output is an AC component, but filtering removes the AC component as it extracts the DC component. Filtering removes it because the AC component must average to zero, and filtering is averaging. Therefore, the RMS value of the output of an RMS converter cannot be measured with an average responding voltmeter, any more than the RMS value of an arbitrary waveform can be measured with an average responding voltmeter to begin with.
The third aspect of interest is not difficult. There frequently is an input DC blocking capacitor ahead of the RMS converter. Its capacitive reactance at low frequencies combines with other impedances in the input circuit to produce a high pass filter. This limits the low frequency response of the RMS converter. Beyond a certain point larger capacitors are not an attractive solution to eliminating the effect of this filtering, and yet it is not always possible to dispense with the blocking capacitor.
Finally, consider the presence of an auto-ranging mechanism in the voltmeter. Typically, the decision about ranging is made using the output of the same RMS converter as is used to supply data to the display. This is generally appropriate in that the ranging decision is being based on an RMS value obtained in accordance with the same general rules as used to get the displayed answer, so that issues such as crest factor and low frequency tracking do not operate one way for the auto-ranging decision and another for the actual displayed reading. Such dissimilar operation could result in inappropriate range selection. However, when long time constants are involved the use of the same RMS converter for both functions can cause long setup times for selecting the proper range. It would be desirable if a separate RMS converter could be employed to steer the auto-ranging mechanism. This additional converter would be in all respects as similar as possible to the main RMS converter (to minimize the ill effects arising from dissimilar response) except that is has a somewhat shorter settling time. The idea is that the auto-ranging mechanism needs a far less accurate measurement, since the logic for auto-ranging incorporates some amount of hysteresis, anyway. The disadvantage of this arrangement is, of course, the additional cost of a whole extra RMS converter.